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Introd uc tion to Finite Elem ent Ana lysisDesign, Analysis, and Simul ation
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Or lots of trian les? Introduction Introduction Introduction FEA Process FEA Process FEA Process An Example An Example An Example Complicated Complicated Complicated Models Models Models Choosing
= .00439 = 2558 psi
Elements Interpreting Interpreting Interpreting Results Results Results FEM Codes FEM Codes FEM Codes
Still too stiff
Still low Wrapping Up Wrapping Up Wrapping Up 4-node triangles are too stiff you can use them here and
there in a model, but you dont want a whole model made from them.
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,preferentially take the load.
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Introduction Introduction Introduction FEA Process FEA Process FEA Process
How about bad elements?An Example An Example An Example Complicated Complicated Complicated Models Models Models Choosing
=
Elements Interpreting Interpreting Interpreting Results Results Results FEM Codes FEM Codes FEM Codes Bad, but not too bad .
= 2194 psiWrapping Up Wrapping Up Wrapping Up
Same
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How about bad elements with QUADRs?
Introduction Introduction Introduction FEA Process FEA Process FEA Process An Example An Example An Example Complicated Complicated Complicated Models Models Models Choosing
= .00481 = 2937 psi
Elements Interpreting Interpreting Interpreting Results Results Results FEM Codes FEM Codes FEM Codes
Much better
Very much better
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QUADR elements tolerate distortions much better then QUAD4s
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What about solid elements? Introduction Introduction Introduction FEA Process FEA Process FEA Process An Example An Example An Example Complicated Complicated Complicated Models Models Models Choosing
= .00499 = 2479 psi
Elements Interpreting Interpreting Interpreting Results Results Results FEM Codes FEM Codes FEM Codes
Great for one elment!
Not too bad here either Wrapping Up Wrapping Up Wrapping Up
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Or lots of solid elements Introduction Introduction Introduction FEA Process FEA Process FEA Process An Example An Example An Example Complicated Complicated Complicated Models Models Models Choosing
= .00501
Elements Interpreting Interpreting Interpreting Results Results Results FEM Codes FEM Codes FEM Codes
Very nice
= 3006 psi Wrapping Up Wrapping Up Wrapping Up Very nice too.
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Introduction Introduction Introduction FEA Process FEA Process FEA Process Why does everyone tell me to avoid TET4 elements? An Example An Example An Example
Complicated Complicated Complicated Models Models Models Choosing
= .00127Elements Interpreting Interpreting Interpreting Results Results Results FEM Codes FEM Codes FEM Codes
As bad as triangles
Recall that TET elements, like triangles, have different formulations
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Lots of TET elements? Introduction Introduction Introduction FEA Process FEA Process FEA Process An Example An Example An Example Complicated Complicated Complicated Models Models Models Choosing
= .00385
Elements Interpreting Interpreting Interpreting Results Results Results FEM Codes FEM Codes FEM Codes
= 2330 psi
Wrapping Up Wrapping Up Wrapping Up Still bad
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Introduction Introduction Introduction FEA Process FEA Process FEA Process
But what if I use parabolic TET10 elements? An Example An Example An Example Complicated Complicated Complicated Models Models Models Choosing
= .00490
Elements Interpreting Interpreting Interpreting Results Results Results FEM Codes FEM Codes FEM Codes
Not bad for a single element!
= 2403 psiMost meshers automatically create these instead of TET4 elements!
Wrapping Up Wrapping Up Wrapping Up Not great, but not terrible
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s ng e e emen s goo or sp acemen , u more are nee e or s ress
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And with lots of them Introduction Introduction Introduction FEA Process FEA Process FEA Process An Example An Example An Example Complicated Complicated Complicated Models Models Models Choosing
= .00501=
Elements Interpreting Interpreting Interpreting Results Results Results FEM Codes FEM Codes FEM Codes
This is a very good answer, but it required 4129 grids!!!
Wrapping Up Wrapping Up Wrapping Up
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A fine mesh cures many ills Introduction Introduction Introduction FEA Process FEA Process FEA Process
It takes a lot of tets to get the right answer Parabolic tets can help a lot, especially for
An Example An Example An Example Complicated Complicated Complicated Models Models Models Choosing
distorted geometries and places where youare forced into tets
Elements Interpreting Interpreting Interpreting Results Results Results FEM Codes FEM Codes FEM Codes
,
but still give bad stresses Its ust hard to et ood stresses be
Wrapping Up Wrapping Up Wrapping Up
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One last conclusion: Introduction Introduction Introduction FEA Process FEA Process FEA Process The structure looked like a beam The beam model ave the best
An Example An Example An Example Complicated Complicated Complicated Models Models Models Choosing Choosing Choosing
answers with the fewest numberof elements
Elements Elements Elements Interpreting Results FEM Codes FEM Codes FEM Codes Wrapping Up Wrapping Up Wrapping Up
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Now lets revisit step 7, getting Introduction Introduction Introduction FEA Process FEA Process FEA Process results FEM codes really only calculate Gauss
An Example An Example An Example Complicated Complicated Complicated Models Models Models Choosing Choosing Choosing
Nastran has no mechanism to get the GaussPoint stresses
Elements Elements Elements Interpreting Results FEM Codes FEM Codes FEM Codes
Nastran interpolates to get centroid values
Nastran extrapolates to get element cornerresults
Wrapping Up Wrapping Up Wrapping Up
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calculated by each element for its own nodes
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Post-Processors plot nodal results Introduction Introduction Introduction FEA Process FEA Process FEA Process The extrapolation usually produces
different values from different
An Example An Example An Example Complicated Complicated Complicated Models Models Models Choosing Choosing Choosing
elements for the same nodes Element results therefore must be
Elements Elements Elements Interpreting Results FEM Codes FEM Codes FEM Codes
single value for the plot Consider this example
Wrapping Up Wrapping Up Wrapping Up
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This is a t ical element Introduction Introduction Introduction FEA Process FEA Process FEA Process N4 N3
G4 G3
An Example An Example An Example Complicated Complicated Complicated Models Models Models Choosing Choosing Choosing
Centroid Elements Elements Elements Interpreting Results FEM Codes FEM Codes FEM Codes
N1 N2
G2 Wrapping Up Wrapping Up Wrapping Up
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Results are re orted at element Introduction Introduction Introduction FEA Process FEA Process FEA Process corners and centroids like this:
100 90
An Example An Example An Example Complicated Complicated Complicated Models Models Models Choosing Choosing Choosing
80
G4 G3 Elements Elements Elements Interpreting Results FEM Codes FEM Codes FEM Codes
G1 G2
Wrapping Up Wrapping Up Wrapping Up
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Consider another element connected Introduction Introduction Introduction FEA Process FEA Process FEA Process to this element:
100 90 110 125
An Example An Example An Example Complicated Complicated Complicated Models Models Models Choosing Choosing Choosing
80
G4 G3
95
G4 G3 Elements Elements Elements Interpreting Results FEM Codes FEM Codes FEM Codes
G1 G2G1 G2
Wrapping Up Wrapping Up Wrapping Up
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And at the connected nodes, the Introduction Introduction Introduction FEA Process FEA Process FEA Process elements have differentanswers!
An Example An Example An Example Complicated Complicated Complicated Models Models Models Choosing Choosing Choosing
This is typical (and not necessarilywrong)
Elements Elements Elements Interpreting Results FEM Codes FEM Codes FEM Codes
processor, it is necessary to comeup with some best guess
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Some t ical wa s to do this: Introduction Introduction Introduction FEA Process FEA Process FEA Process Average the connected elements
110+100/2 = 105
An Example An Example An Example Complicated Complicated Complicated Models Models Models Choosing Choosing Choosing
Most common default method Stress Averaging
Elements Elements Elements Interpreting Results FEM Codes FEM Codes FEM Codes
Pick the largest Max(110,100) = 110
Wrapping Up Wrapping Up Wrapping Up
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This is fine as long as the elements Introduction Introduction Introduction FEA Process FEA Process FEA Process are the same thickness and that theresults are something that makes
An Example An Example An Example Complicated Complicated Complicated Models Models Models Choosing Choosing Choosing
v x But if they are different?
Elements Elements Elements Interpreting Results FEM Codes FEM Codes FEM Codes
Wrapping Up Wrapping Up Wrapping Up
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A beam made from plates of twoIntroduction Introduction Introduction FEA Process FEA Process FEA Process
t c nesses: An Example An Example An Example Complicated Complicated Complicated Models Models Models Choosing Choosing Choosing
Elements Elements Elements Interpreting Results FEM Codes FEM Codes FEM Codes
this:
Wrapping Up Wrapping Up Wrapping Up
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The moral of the stor isIntroduction Introduction Introduction FEA Process FEA Process FEA Process
Be careful with what you plot Smooth contours look nice but may
An Example An Example An Example Complicated Complicated Complicated Models Models Models Choosing Choosing Choosing
hide high stresses Elements Elements Elements Interpreting Results FEM Codes FEM Codes FEM Codes Wrapping Up Wrapping Up Wrapping Up
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Look at the real element results! Dont avera e ad acent elements es eciall if the
Introduction Introduction Introduction FEA Process FEA Process FEA Process
are different thicknesses or materials An Example An Example An Example Complicated Complicated Complicated Models Models Models Choosing Choosing Choosing
Elements Elements Elements Interpreting Results FEM Codes FEM Codes FEM Codes Wrapping Up Wrapping Up Wrapping Up
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Not all FEM Codes are created equallyIntroduction Introduction Introduction FEA Process FEA Process FEA Process
Linear codes optimized for infinitesimal
displacements
An Example An Example An Example Complicated Complicated Complicated Models Models Models Choosing Choosing Choosing
[K] changes during solution Implicit codes statics, build and solve [K] -1
Elements Elements Elements Interpreting Interpreting Interpreting Results Results Results FEM Codes , e ,
Eulerian codes for fluids (constant volume) Thermal codes heat transfer
Wrapping Up Wrapping Up Wrapping Up
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Infinitesimal displacementsIntroduction Introduction Introduction FEA Process FEA Process FEA Process
e mo on o e no e sinconsequential to the solution of theproblem
An Example An Example An Example Complicated Complicated Complicated Models Models Models Choosing Choosing Choosing
Large displacements The motion of the nodes changes the
Elements Elements Elements Interpreting Interpreting Interpreting Results Results Results FEM Codes
so u on
Example fixed-fixed beam Bouncing ball
Wrapping Up Wrapping Up Wrapping Up
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Cantilever beamIntroduction Introduction Introduction FEA Process FEA Process FEA Process An Example An Example An Example Complicated Complicated Complicated Models Models Models Choosing Choosing Choosing
Tip motion does not affect root loadsignificantly for small displacements
Elements Elements Elements Interpreting Interpreting Interpreting Results Results Results FEM Codes
ect s proport ona to s n t eta
=~ 0 for small deflections (
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Fixed-Fixed Beam Introduction Introduction Introduction
FEA Process FEA Process FEA Process An Example An Example An Example Complicated Complicated Complicated Models Models Models Choosing Choosing Choosing
Initially no bending, no axial load all shear Once center deflects, bending and axial loads
appear and grow
Elements Elements Elements Interpreting Interpreting Interpreting Results Results Results FEM Codes
Becomes primary load path!
Motion of the beam center affects the responsessignificantly
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An ID code will not show this effect!
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Linear Most FEM codesIntroduction Introduction Introduction FEA Process FEA Process FEA Process
including NEi Nastran do this) Stiffness matrix sta s the same for
An Example An Example An Example Complicated Complicated Complicated Models Models Models Choosing Choosing Choosing
the entire solution Single unique static solution
Elements Elements Elements Interpreting Interpreting Interpreting Results Results Results FEM Codes
Loads can be applied in any order
Unique steps for other solutions
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, ,
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Implicit (NEi Nastran or NEi Fusion)Introduction Introduction Introduction FEA Process FEA Process FEA Process
u s ness an mass ma r ces assem e ansolved
Variety of solutions
An Example An Example An Example Complicated Complicated Complicated Models Models Models Choosing Choosing Choosing
, ,
Explicit (NEI Explicit for example) Solved at element level no big matrices
Elements Elements Elements Interpreting Interpreting Interpreting Results Results Results FEM Codes
n y trans ent so ut on ava a e
Good for highly nonlinear problems Good for very large models
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In general, the finite element method Introduction Introduction Introduction
FEA Process FEA Process FEA Process described by partial differentialequations
An Example An Example An Example Complicated Complicated Complicated Models Models Models Choosing Choosing Choosing
Thermal Analysis
Fluid flow/wave propagation
Elements Elements Elements Interpreting Interpreting Interpreting Results Results Results FEM Codes FEM Codes FEM Codes
Electromagnetic
Dynamics
Wrapping Up
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Example: Steady-state heat conductionIntroduction Introduction Introduction FEA Process FEA Process FEA Process
Replace the structural stiffness matrix withthe matrix of thermal conductivities
An Example An Example An Example Complicated Complicated Complicated Models Models Models Choosing Choosing Choosing
= ,conductivity and T is temperature
Single DOF at each node (temperature)
Elements Elements Elements Interpreting Interpreting Interpreting Results Results Results Wrapping Up
This is why most FEM codes also do thermalproblems!
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K. J. Bathe
Finite Element Procedures in Engineering AnalysisIntroduction Introduction Introduction FEA Process FEA Process FEA Process
ren ce- a , R. D. Cook
Concepts and Applications of Finite Element AnalysisJohn Wiley & Sons, 1989
An Example An Example An Example Complicated Complicated Complicated Models Models Models Choosing Choosing Choosing
Harry G. ShaefferMSC/NASTRAN PRIMER, 1998
Richard H. MacNeal
Elements Elements Elements Interpreting Interpreting Interpreting Results Results Results Wrapping Up
O. C. Zienkiewicz
The Finite Element MethodMcGraw-Hill, 1994
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An Introduction to the Finite Element MethodMcGraw-Hill, 1984
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Now you should be somewhatIntroduction Introduction Introduction FEA Process FEA Process FEA Process
familiar with some FEM buzzwords: Gauss Points
An Example An Example An Example Complicated Complicated Complicated Models Models Models Choosing Choosing Choosing
Shape Functions Continuity
Elements Elements Elements Interpreting Interpreting Interpreting Results Results Results Wrapping Up
Large Displacements
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